THE CRITICAL CASE OF 4TH-ORDER RESONANCE IN A HAMILTONIAN SYSTEM WITHONE DEGREE-OF-FREEDOM

The motion of a time-periodic Hamiltonian system with one degree of fr eedom in the neighbourhood of an equilibrium position is studied. It i s assumed that the equilibrium is stable in the first approximation an d that fourth-order resonance is present. The critical case is conside red, when the system parameters are such that, in order to draw rigoro us conclusions about the stability of the equilibrium, terms of order higher than four in the series expansion of the Hamiltonian must be ta ken into account. Sufficient conditions are derived for stability and instability, and the bifurcations of periodic motions are investigated in the neighbourhood of the equilibrium position when the system para meters pass through values corresponding to the critical case. The res ults are applied in the problem of the motion of a sphere in a uniform gravity field when there are collisions with the surface of an ellipt ic cylinder with a horizontal generator. (C) 1997 Elsevier Science Ltd . All rights reserved.